How to know the radius of curvature that can be achieve with a line?

Question

I have a line, of x length. I want to have a semicircle with that length. How can I know the max and min radius I can have with that line?

Answer 1

It is nice to meet you.

Seeing as the perimeter of a semicircle=P=P(semicircle)+P(base)=$\frac{2πr}{2}+2r$:

This can be simplified down to P=πr+2r=r(π+2). Because the perimeter, or length, of the semicircle to be the same length as a line of length x, the following is needed:

P=r(π+2)=x⇔$r=\frac{x}{π+2}$.

Please correct me if I am wrong.

Here is an article found after this answer was posted: https://socratic.org/questions/what-is-the-formula-for-the-radius-of-a-semi-circle.

The max radius is the min radius in flat euclidean space, but if there was a transformation in the space, perhaps a linear transformation or a hyperbolic space was introduced, this could deform the semicircle just as a line is deformed in a line integral over another surface.

However, that is not an area of study that I am used to.

Example?: Construction of Hyperbolic Circles With a Given Radius